XFEM for reinforced concrete

Aalborg University Structural and Civil Engineering, 10th SemesterSchool of Engineering and Science Studyboard of Civil Engineering Sohngrdsholmsvej 57 www.bsn.aau.dk

Synopsis:

Title:

Numerical analysis of a Reinforced Concrete Beam in Abaqus 6.10 B10K, spring 2011 B122b

Project period: Project group:

Baldvin Johannsson

This report deals with the modeling of cracks in a three-dimensional reinforced concrete beam subjected to three point bending. The commercial nite element program Abaqus 6.10 is used to model the beam. In Abaqus 6.10 the eXtended Finite Element Method, abbreviated XFEM, is used to model the cracking in cooperation with the Concrete Damaged Plasticity material model, abbreviated CDP, to model the tensile and compressive behavior. Prior to modeling the three-dimensional beam, benchmark tests are performed to validate the quality of the implementation of the XFEM and the CDP material model. Load-deection curves are analyzed for two numerical models; with and without the XFEM, and compared to analytical load-deection relations from EN 1992-1-1 [2004]. The cross-sectional stress distribution is plotted for various stages on the loaddeection curves for the numerical models and compared to a theoretical cross-sectional stress distribution at the corresponding stages. Crack widths, spacing and patterns are analyzed and compared to expressions given in EN 1992-1-1 [2004]. This report has shown that it is possible to combine the CDP material model with the XFEM in order to visualize cracks in reinforced concrete beams. In general, good agreement is found between analytical and numerical results. However, it is concluded that improvements are necessary to the implementation of the XFEM in Abaqus 6.10.

Poul Reitzel

Supervisors: Christian Frier Print runs: 10 Number of pages: Appendix: Completed:

The report's content is freely available, but the publication (with source indications) may only happen by agreement with the authors.

PrefaceThis report is a product of Poul Reitzel's and Baldvin Johannsson's project work at the 4th semester of the master degree in Structural and Civil Engineering at Aalborg University. The project has been completed within the period of 1st of February, 2011, to the 10th of June, 2011, under the supervision of Christian Frier. The report is prepared and made in accordance and compliance with the current curriculum of the 4th semester of the master programme in Civil Engineering at Aalborg University, Denmark. The project is based on the theme "Numerical analysis of a Reinforced Concrete Beam in Abaqus 6.10". The project aims at the increase of knowledge to apply an advanced computational method for the evaluation of crack formation and propagation in reinforced concrete. The project report consists of two parts, the main project and the appendix. A reference to the appendix can be: Appendix B. The main project examines the use of the extended nite element method, abbreviated XFEM, in combination with the Concrete Damaged Plasticity material model, abbreviated CDP, in Abaqus 6.10 to model a three-dimensional reinforced concrete beam loaded to failure. The project report uses the Harvard method of bibliography with the name of the source author and year of publication inserted in brackets after the text, for example: Irwin [1958]. The list of all references is found in the bibliography at the end of the report. The les used in Matlab and Abaqus 6.10 can be found on the attached CD. An introduction to the les and how they work can be found in Appendix B. The authors expect the reader to have a basic knowledge of the standard nite element method. Experience with computational engineering within the nite element method framework will make the interpretation of certain technical terms easier.

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Table of contentsChapter 1 Introduction Chapter 2 Fracture mechanics for concrete2.1 2.2 2.3 3.1 3.2 4.1 4.2

3 7

The fracture process in concrete . . . . . . . . . . . . . . . . . . . . . . . . . 8 Linear elastic fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . 10 Non-linear fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Smeared crack concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Discrete crack concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Smeared vs. discrete crack modelling

17

Chapter 4 The eXtended Finite Element MethodDiscontinuities and high gradients . . . Level-set method . . . . . . . . . . . . . 4.2.1 Description of closed interfaces . 4.2.2 Description of open interfaces . . General formulation of the XFEM . . . Choice of enriched nodes . . . . . . . . . Global enrichment functions . . . . . . . 4.5.1 Weak discontinuities . . . . . . . 4.5.2 Strong discontinuities . . . . . . 4.5.3 Singularities . . . . . . . . . . . . Cohesive segments method . . . . . . . . Phantom-node method . . . . . . . . . . Numerical integration of the weak form Governing equations . . . . . . . . . . .

23

4.3 4.4 4.5

4.6 4.7 4.8 4.9

24 25 27 28 29 31 33 34 35 35 36 37 39 41

Chapter 5 Discontinuous modeling in Abaqus Chapter 6 Verication of the XFEM and Abaqus6.1 6.2 6.3 6.4 6.5 7.1 Crack-hole interaction . . . . . . . . . . . . . . . . . . . . . . Verication of the CDP material model . . . . . . . . . . . . Crack propagation in a concrete beam in three point bending Crack formation analysis of a reinforced concrete plate . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 5354 56 57 62 65

Chapter 7 Results and discussion of 3d beam analysis

Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 v

67

Group B122b - Spring 20117.1.1 Boundary conditions . . . . . . . 7.1.2 Input for the material model . . 7.1.3 Reinforcement Model . . . . . . List of studies . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . 7.3.1 Load-deection curves . . . . . . 7.3.2 Cross-sectional stress distribution 7.3.3 Crack widths and spacing . . . . 7.3.4 Crack pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

TABLE OF CONTENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 70 72 74 74 74 79 84 87

7.2 7.3

Chapter 8 Conclusion Chapter 9 Suggestions for future work Appendix A Concrete Damaged Plasticity material modelA.1 Concrete Damaged Plasticity . . . . . . . . . . . . . . . . . A.1.1 Stress-strain relations . . . . . . . . . . . . . . . . . A.2 Yield function . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Damage and stiness degradation for unixial loading A.2.2 Flow rule . . . . . . . . . . . . . . . . . . . . . . . . B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 B.9 B.10 B.11 B.12 B.13 B.14 B.15 B.16 B.17 B.18 B.19 B.20 B.21 B.22 B.23 B.24 B.25 DVD 1 . . . . . . . . . . . . . . . . . . . . . . . . 3D-Beam XFEM.cae . . . . . . . . . . . . . . . . Maximum and minimum reinforcement ratio.pdf Crack widths and spacing.pdf . . . . . . . . . . . Shear deformation.pdf . . . . . . . . . . . . . . . Ultimate load and corresponding deection.pdf . Seed45XFEM.odb . . . . . . . . . . . . . . . . . Seed50XFEM.odb . . . . . . . . . . . . . . . . . Seed60XFEM.odb . . . . . . . . . . . . . . . . . Seed70XFEM.odb . . . . . . . . . . . . . . . . . Seed80XFEM.odb . . . . . . . . . . . . . . . . . Seed150XFEM.odb . . . . . . . . . . . . . . . . . DVD 2 . . . . . . . . . . . . . . . . . . . . . . . . 3D-Beam CDP.cae . . . . . . . . . . . . . . . . . Seed45CDPFull.odb . . . . . . . . . . . . . . . . Seed50CDPFull.odb . . . . . . . . . . . . . . . . Seed60CDPFull.odb . . . . . . . . . . . . . . . . Seed70CDPFull.odb . . . . . . . . . . . . . . . . Seed80CDPFull.odb . . . . . . . . . . . . . . . . Seed150CDPFull.odb . . . . . . . . . . . . . . . . Seed50CDPRed.odb . . . . . . . . . . . . . . . . Seed60CDPRed.odb . . . . . . . . . . . . . . . . Seed70CDPRed.odb . . . . . . . . . . . . . . . . Seed80CDPRed.odb . . . . . . . . . . . . . . . . Seed150CDPRed.odb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 93 33 3 5 7 8

Appendix B Guide to Appendix CD

9 9 9 9 9 10 10 10 10 10 10 10 11 11 11 11 11 11 11 12 12 12 12 12 12

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vi

TABLE OF CONTENTS

Master Thesis

Bibliography

13

vii

ResumP nuvrende tidspunkt er der ingen tilgngelige resultater for en fuldstndig analyse af revnedannelse i armerede betonkonstruktioner i Abaqus 6.10, der ved hjlp af den skaldte "eXtended Finite Element Method", forkortet XFEM, er i stand til at forudsige revnemnstre og revnevidder. Samtidig tilbyder Abaqus 6.10 muligheden for, at modellere udviklingen i stivheden for konstruktioner hele vejen frem til brud ved hjlp af materialemodellen Concrete Damaged Plasticity, forkortet CDP. Modeller, der kan forudsige revnevkst, revnemnstre og revnevidder i armerede betonkonstruktioner, er pkrvede i henhold til at overholde krav til revnevidder og revneafstande fra gldende normer. En mere prcis analyse af revnevidder og revneafstande er motivationen bag den forestende rapport. Analyse af konstruktioner med komplekse geometrier vil ofte krve anvendelse af numeriske vrktjer som fx. nite element metoden. Denne afhandling omhandler kohsiv revnevkst i beton inden for rammerne af XFEM. Hovedparten af arbejdet er relateret til modellering af revnevkst i en tre-dimensionel armeret betonbjlke ud fra et modelleringsteknisk perspektiv. XFEM er baseret p berigelse af ytningsfeltet i elementer, der er gennemskret af en diskontinuitet. Konceptet for berigelsen er baseret p lokale enhedsytningsfelter (partition of unity). Den anvendte berigelse er elementlokal, det vil sige kun elementer der er gennemskret af diskontinuiteten berres af berigelsen. Modelleringsarbejdet i rapporten tager udgangspunkt i en tre-dimensionel armeret betonbjlke udsat for trepunkts bjning og anvendelsen af materialemodellen CDP. Det undersges hvorvidt aktiveringen af XFEM forbedrer resultaterne med hensyn til last-ytningskurver, spndingsfordelinger i et revnet tvrsnit samt revnedannelse. Resultaterne sammenlignes med retningslinjer og formler fra gldende normer i EN 1992-1-1 [2004]. Gode resultater er opnet med hensyn til svel forudsigelse af den fulde last-ytningsrespons og spndingsfordelinger i tvrsnittet. Last-ytningskurverne viste sig, at vre uafhngige af aktiveringen af XFEM, hvorimod aktiveringen af XFEM havde positiv indydelse p spndingerne i tvrsnittet. Resultaterne for revnevidder, revneafstand og revnemnster er kun opnelig ved aktiveringen af XFEM. Resultaterne viste, at en forbedring af den nuvrende implementering af XFEM i Abaqus 6.10 er ndvendig p grund af antagelser vedrrende propagering og initiering af revner.

1

Introduction

1

In this chapter the motivation for this project is described followed by a presentation of the problem to be handled. This leads to the problem formulation of the project, covered in the report. Finally, a list of objectives is stated to answer the problem formulation.Concrete is a stonelike material obtained by permitting a mixture of cement, sand and gravel or other aggregate and water, to harden in forms of the desired shape of the structure. Additives like silica fume and y ash can be added to the concrete mixture for increased strength and workability. Concrete has become a popular material in civil engineering for several reasons, such as the low cost of the aggregate, the accessibility of the needed materials and its high compressive strength, which makes it suitable for members primarily in compression such as columns. For example the pillars of the wellknown Storeblt bridge is constructed of reinforced concrete, see gure 1.1.

Figure 1.1.

The Storeblt bridge. Zeelandsite [2011]

On the other hand concrete is a relatively brittle material with low tensile strength compared to the compressive strength. For this reason, concrete elements are reinforced 3

Group B122b - Spring 2011

1. Introduction

with steel bars in areas of the cross-section which is subjected to tension to provide some tensile resistance. In this way, reinforced concrete acts a bi-material utilizing the high compressive strength of concrete and the tensile strength of steel. A.H. Nilson, D. Darwin and Charles W. Dolan [2004] Methods of determining the ultimate strength of concrete based on linear elasticity or plasticity is widely developed. However, the possibilities for analysis in the serviceability limit state, e.g. deection and crack width estimations, are lacking and empirical in EN 1992-1-1 [2004]. Moreover, the proposed deection formula in EN 1992-1-1 [2004] does not take reinforcement arrangement or geometry into account. The last statement is also true for the estimation of crack width and spacing in EN 1992-1-1 [2004]. For structures with a complex geometry the Finite Element Method, abbreviated FEM, must be used, which should be able to model the cracking of the concrete. Within the Finite Element framework it is desirable to represent the actual stress distribution and the development hereof, e.g. by employing an advanced material model, to precisely model the section stresses at every load stage. A principle sketch of the stress development in a cross-section for increasing bending moment is shown in gure 1.2.

Figure 1.2.

Principle sketch of the stress development for increasing bending moment. Sren Madsen [2009]

Cracking of the concrete is unavoidable due to its low tensile strength and low extensibility. The downside of cracking is twofold. Cracking exposes the reinforcing steel to the surrounding environment, which can cause corrosion of the steel. Secondly, a cracked construction loses its aesthetical qualities and appears unsafe to reside in. Studies performed by Arya and Ofori-Darko [1996] reveal that crack spacing is a governing factor in the rate of corrosion. They found, that several small cracks are more severe than one big crack. Other studies performed by Schiel and Raupach [1997] and Mohammed et al. [2001] show, that the crack width inuences the time to initiation of the corrosion. To avoid this, the engineer must ensure that the crack widths and spacing are within the allowable limits put forth by the governing code. In addition, cracking of a member will cause reduction in bending stiness, which inuences the deection of the member. It is important to accurately assess the inuence of cracks on the deection of concrete members in the serviceability limit state. Piyasena [2002] The standard FEM is well established as a robust and reliable numerical technique for studying the behavior of a wide range of engineering and physical problems. All physical phenomena encountered in engineering mechanics are modeled by dierential equations, usually too complicated to be solved by analytical methods. For problems 4

Master Thesiswhere the solution variables behave in a continuous manner, the FEM is a highly suited method for approximating the solution to the dierential equation governing the addressed problem. However, a large number of models in continuum mechanics involve solutions that are discontinuous in local parts of the solution domain. For example, displacements change discontinuously across cracks. In the FEM special care must be taken in the construction of an appropriate mesh, as element topology must align with the geometry of the discontinuity, and this is not desirable in applications, where the crack location is unknown a-priori. In applications, such as crack analysis, where the FEM encounters problems, other, more appropriate numerical techniques exist. One class of techniques is the so-called enriched methods, which are advantageous for problems having non-smooth and non-regular solution characteristics. A popular enriched method is the so-called extended nite element method, abbreviated XFEM. The XFEM was implemented by Dassault Systmes Simulia Corp. [2010] in their latest version of Abaqus (6.10), which puts the engineer in a position of being able to qualitatively estimate crack patterns, spacing and widths for arbitrary geometries. Abaqus 6.10 also oers advanced material models to accurately model the behavior of concrete shown in gure 1.2. Thomas-Peter Fries and Andreas Zilian. [2010] This report approaches the problem of modeling a three-dimensional reinforced concrete beam loaded to failure. The report aims at identifying the location of crack initialization and propagation independent of the mesh, and estimating the crack widths and spacing. Moreover, it is desired to reect the damage process of crushing and cracking in the cross-sectional stress distribution. The above introduction leads to the following specic problem formulation for this project:

Application of the extended nite element method and the Concrete Damaged Plasticity material model for cracking simulation in a three-dimensional reinforced concrete beam using the commercial nite element program Abaqus.In order to handle the described problem, the following objectives for the project are set: Review of linear and non-linear fracture mechanics for concrete. Review of available material models for modeling of cracks in concrete. Introduction to the XFEM with focus on crack modeling. Description of the implementation of the XFEM in Abaqus 6.10. Examination of the quality of the implementation of the Concrete Damaged Plasticity material model and the XFEM in Abaqus 6.10, by performing four benchmark tests. Review of existing guidelines with respect to crack width and spacing in EN 1992-1-1 [2004]. Analysis of load-deection curves and stress distributions using the Concrete Damaged Plasticity material model, with and without the XFEM. Analysis of crack width, crack spacing and crack pattern using the Concrete Damaged Plasticity material model and the XFEM. 5


1. Introduction

The boundary conditions, dimensions and reinforcement arrangement in the threedimensional reinforced concrete beam are shown in gure 1.3. This specic beam is analyzed, because the dimensions and material properties are typical for construction beams.

Figure 1.3.

Boundary conditions, dimensions and reinforcement arrangement in the examined reinforced concrete beam.

Note, that shear reinforcement is not incorporated into the model in order to simplify the modeling work load in Abaqus 6.10.

6

Fracture mechanics for concrete

2

This chapter presents an introduction to cracking in concrete with focus on the fracture process in compression and tension. This is followed by a description of the fundamental concepts of linear elastic fracture mechanics, LEFM. Within the scope of LEFM the theory in Grith [1921]/Irwin [1958] is presented along with the cohesive crack model proposed by Dugdale [1960] and Barrenblatt [1959]. Following the description of LEFM, the work done by A. Hillerborg and Peterson [1976], within the eld of nonlinear fracture mechanics, is presented.Concrete is a heterogeneous anisotropic non-linear inelastic composite material, which is full of aws that may initiate crack growth when the concrete is subjected to stress. Failure of concrete typically involves growth of large cracking zones and the formation of large cracks before the maximum load is reached. This fact, and several properties of concrete, points toward the use of fracture mechanics. Furthermore, the tensile strength of concrete is neglected in most serviceability and limit state calculations. Neglecting the tensile strength of concrete makes it dicult to interpret the eect of cracking in concrete. This may be accounted for by applying a fracture mechanics approach. Five arguments stated by the ACI Committee [1992] suggests why fracture mechanics should be adopted into certain aspects of design of concrete structures: 1. It is not sucient to specify how cracking is initiated, e.g. by a stress criterion, but also how it will propagate. The growth of a crack requires the consumption of a certain amount of energy, called the fracture energy. Therefore, crack propagation can only be studied through an energy criterion. 2. The calculations must be objective, i.e. mesh renement, choice of coordinates etc. must not aect the results. This entails that the energy dissipated through cracking is constant, which is done by specifying the energy dissipated per unit length of the crack. 3. Two basic types of structural failure may be stated: brittle and plastic. Plastic failure occurs in materials with a long yield plateau and the structure develops plastic hinges. For materials with a lack of yield plateau, the fracture is brittle, 7


2. Fracture mechanics for concrete

which implies the existence of softening. During softening the failure zone propagates throughout the structure, so the failure is propagating. 4. The area under the load-displacement curve determines the amount of energy consumed during failure process. This energy determines the ductility of the structure, and a limit state analysis cannot give an indication of this, because the post-peak response is not taken into account. 5. Fracture mechanics may opposite to strength criterions predict the inuence of the structural size on the failure load and ductility. The ve arguments stated above motivate towards using fracture mechanics in the modeling of concrete when cracking is of interest. Thus fracture mechanics may lead to a physical explanation of cracking in concrete, that the current codes, e.g. EN 1992-1-1 [2004], do not by their present empirical formulas.

2.1 The fracture process in concreteIn 1983 Wittmann [1983] suggested to dierentiate between three dierent levels of cracking in concrete. The levels are categorized as follows: Micro cracks that can only be observed by an electron microscope. Meso cracks that can be observed using a conventional microscope. Macro cracks that visible to the naked eye. Micro cracks occur on the level of the hydrated cement, where cracks form in the cement paste. Meso cracks form in the bond between aggregates and the cement paste. Finally, macro cracks form in the mortar between the aggregates.

The fracture process in compressionThe compressive stress-strain curve for concrete can be divided into four regions, see gure 2.1. The gure describes four dierent states of compressive cracking.

8

2.1. The fracture process in concrete

Master Thesis

Figure 2.1.

The compressive stress-strain curve for concrete. The curve is divided into four regions for dierent states of cracking. J.P. Ulfkjr [1992]

Initial cracks on the micro-level, caused by shrinkage, swelling and bleeding, are observed in the cement paste prior to loading. For loads of approximately 0 30 % of the ultimate load the stress-stain curve is approximately linear and no growth of the initial cracks is observed. Between approximately 30 50 % of the ultimate load a growth in bonding cracks between the cement paste and aggregates is observed. The cement paste and the aggregates have dierent elastic modulii, which increases the non-linearity of the stressstrain curve. Beyond 50 % of the ultimate load macro-cracks start to slowly form in the mortar, running between the aggregates parallel with the load direction. At app. 75 % of the ultimate load a more complex crack formation is established, where the bonding cracks and the cracks in the mortar coalesce until nally failure occurs. J.P. Ulfkjr [1992]

The fracture process in tensionThe tensile strength of concrete is, much like the compressive strength, dependent on the strength of each link in the cracking process, i.e micro-cracks in the cement paste, meso-cracks in the bond and macro-cracks in the mortar. Consider a concrete rod under pure tensile loading, see gure 2.2. The fracture process initiates with crack growth of existing micro cracks at approximately 80 % of the ultimate tensile load. This is followed by formation of new cracks and a halt in formation of others due to stress redistribution and the presence of aggregates in the crack path. These cracks are uniformly distributed throughout the concrete specimen. When the ultimate tensile load is reached, a localized fracture zone will form in which a macro-crack, that splits the specimen in two, will form. The fracture zone develops in the weakest part of the specimen. J.P. Ulfkjr [1992]

9



Figure 2.2.

A concrete rod subjected to pure tensile loading. Outside the fracture zone, the cracks are uniformly distributed. Inside the fracture zone a macro-crack forms which splits the rod in two.

In the following section the basis of linear elastic fracture mechanics is presented.

2.2 Linear elastic fracture mechanicsGrith [1921] was the rst to develop a method of analysis for the description of fracture in brittle materials. Grith found that, due to small aws and cracks, stress concentrations arise under loading, which explains why the theoretical strength is higher than the observed strength of brittle materials. Grith studied the inuence of a sharp crack on an arbitrary body with the thickness t loaded remotely from the crack-tip with an arbitrary load F , see gure 2.3.

Figure 2.3.

Arbitrary body with an internal crack of length a subjected to an arbitrary force, F.

By superposition, the potential energy of the body is given by 2.1.

= e + F + K + c10

(2.1)

2.2. Linear elastic fracture mechanicse F K cThe The The The elastic energy content in the body. potential of the external forces. total kinetic energy in the system. fracture potential.

Master Thesis

The fracture potential, c , is the energy that dissipates during crack growth. By assuming that crack growth is only dependent on the crack length, a, the equilibrium equation can be stated, by requiring that the potential energy of the system equals zero, see 2.2.

=0 t a

(2.2)

Grith [1921] introduced a parameter, the energy release rate, G, and dened a fracture criteria, see equation 2.3.

G=

c =R t a

(2.3)

where R is the fracture resistance of the material, which is assumed to be constant in LEFM. The total potential energy of a system increases when a crack is formed because a new surface is created, thus increasing the fracture potential. However, the formation of a crack consumes an amount of energy, G, in the form of surface energy and frictional energy. If the energy release rate is larger than the energy required to form a crack, see 2.4, crack growth is unstable.

R G > =0 a a

(2.4)

The method proposed by Grith [1921] was based on energy considerations, but is not adequate in design situations. For this reason, Irwin [1958] developed the stress intensity factor, abbreviated SIF, concept. The SIF can be understood as a measure of the strength of a singularity, understood in the sense that the SIF amplies the magnitude of the stresses around the singularity. The literature distinguishes between the three dierent fracture modes shown in gure 2.4.

11



Figure 2.4.

Top: Crack mode I. Middle: Crack mode II. Bottom: Crack mode III. NDT [2011]

Mode I fracture is the condition in which the crack plane is perpendicular to the direction of the applied load and mode II fracture is the condition in which the crack plane is parallel to the direction of the applied load. Mode III fracture corresponds to a tearing mode and is only relevant in three dimensions. Mode I and mode II fracture is also referred to as an opening and in-plane shear mode, respectively. Irwin [1958] showed that the stress variation near a crack tip in a linear elastic material is dependent on the distance to the crack tip, called r. More precisely, the stress is singular at the crack-tip with a square-root singularity in r, see equation 2.5.

K ij = fij () + higher order terms 2r ij K , r fijThe stress tensor. The stress intensity factor. The polar coordinates at the crack-tip. A trigonometric function.

(2.5)

From equation 2.5 it can be seen, that a linear relationship exists between the stress and the SIF, which reects the linear nature of the theory of elasticity. In practical calculations, only the rst order term of equation 2.5 is included. This is because, that for r 0, the rst order term approaches innity while the higher order terms are constant or zero. Because the stress tends towards innity when r 0 a stress criterion as a failure criterion is not appropriate. For this reason, Irwin derived a relationship between the SIF and the energy release rate, G, see 2.6. (2.6) K = GE 12

2.3. Non-linear fracture mechanicsThe fracture criterion can thereby be written as

Master Thesis

K = Kc

(2.7)

It should be noted that the global energy balance criteria by citebib:grith is equivalent to the local stress criteria by Irwin [1958]. Moreover, Kc is also referred to as the fracture toughness of the material, and is regarded as a constant in LEFM. The Grith [1921]/Irwin [1958] theory assumes that the stresses in the vicinity of the crack-tip tend to innity. This contravenes the principle of linear elasticity, relating small strains to stresses through Hooke's law. In the fracture process zone, abbreviated FPZ, ahead of the crack-tip, plastic deformation of the material occurs. Specically for concrete debonding of aggregate from the cement matrix and microcracking occurs. Moreover, cracks coalesce, branch and deect in the FPZ. To describe this highly non-linear phenomenon, non-linear fracture mechanics, abbreviated NLFM, must be adopted.

2.3 Non-linear fracture mechanicsThe rst attempt to analyze plasticity at the crack-tip was done by Dugdale [1960] and Barrenblatt [1959]. Dugdale [1960] and Barrenblatt [1959] independently proposed two models, in which closing forces were included at the crack-tip. The closing forces are also referred to as cohesive forces, and the small zone, over which they act, is termed the cohesive zone. The stress singularity that arises at the crack-tip using the Grith [1921]/Irwin [1958] theory vanishes when the approaches suggested by Dugdale [1960] or Barrenblatt [1959] are used. In the model suggested by Dugdale [1960], the cohesive closure stress is the yield strength of the considered material. In the model suggested by Barrenblatt [1959], the cohesive closure stress is a characteristic material molecular force of cohesion and has a generally unknown variation along the FPZ. The principle of the cohesive zone model by Barrenblatt [1959] is shown in gure 2.5.

Figure 2.5.

The cohesive zone model by Barrenblatt [1959]. A body with a crack of length 2a subjected to tension, . Cohesive stresses, q(x), act along a cohesive zones of length c at each crack-tip.13



Inspired by the concept of Dugdale [1960] and Barrenblatt [1959], A. Hillerborg and Peterson [1976] redened the FPZ by introducing a so-called ctitious crack in front of the real crack-tip. The purpose of introducing a ctitious crack was to improve the description of the tractions acting in FPZ. The closure stress in the FPZ has a maximum value of ft , i.e. the tensile strength of the material, at the boundary of the FPZ and is zero at the tip of the real crack. The variation in-between is given by a softening law, relating stresses to the crack opening displacement, w. Similar to elastic materials with a constitutive law described by e.g. Hooke's law, the tension softening law is the constitutive law in the FPZ. Thus the tension softening law describes the transition between the continuous state and the discontinuous state of the material behavior. Figure 2.6 illustrates a typical tensile load-displacement response of concrete and the related ctitious crack ahead of the real crack. Note that the FPZ extends only over the length of the tension softening region BCD, see gure 2.6. Tension softening is the relationship between the cohesive stress and the crack opening displacement in the FPZ. Note that the relation between the closure stress and the crack opening displacement is non-linear, and that a degradation of the Young's modulus occurs gradually inside the FPZ. J.L. Asferg [2006]

Figure 2.6.

(a) Typical tensile load-displacement curve of concrete with letters indicating the crack-state: A: Uncracked, linear-elastic behavior, B: The tensile strength has been reached and microcracking and tension softening occur, C: Stress bridging D: The crack becomes traction-free. (b) The FPZ related to the load-displacement curve. The variation of the cohesive stresses is indicated and the crack is divided into a traction-free zone, a microcracking/bridging zone and a microcracking zone.

14

2.3. Non-linear fracture mechanics

Master Thesis

As previously mentioned energy is absorbed during crack growth in order to form the new crack surfaces. The amount of absorbed energy is the fracture energy, Gf , which is equal to the area under the tension-softening curve, see gure 2.7 and equation 2.8.

Figure 2.7.

Tension-softening.

wc

Gf =0

(w) dw

(2.8)

As a nal remark, the ctitious crack model proposed by A. Hillerborg and Peterson [1976] assumes, that the FPZ has negligible width. For this reason, the model belongs to the class of discrete crack models. The ctitious crack model by A. Hillerborg and Peterson [1976] is the basis for the cohesive segments method used in Abaqus 6.10. The cohesive segments method is described in chapter 4.

15

Smeared vs. discrete crack modelling

3

This chapter presents the concepts of smeared and discrete crack models for concrete. Popular techniques, e.g. the XFEM, available for discrete crack modeling are discussed. Advantages and drawbacks are identied and pointed out for a discrete vs. smeared approach to crack modeling. The purpose of this chapter is to illustrate the motivation for working with the XFEM in this report.In the late 1960's D. Ngo and A.C. Scordelis [1967] performed a numerical simulation of discrete cracks in concrete. At the same time, Rashid [1968] successfully applied a smeared crack model for concrete. Discrete crack simulation aims at the initiation and propagation of dominating cracks, whereas the smeared crack model is based on the observation, that the heterogeneity of concrete leads to the formation of many, small cracks which, only in a later stage, nucleates to form one larger, dominant crack. The smeared crack model captures the deterioration process by smearing the eect of microcracks, that is, a reduction in stiness, over a given volume. With respect to the problem formulation stated in chapter 1, the objective of the numerical model is to Identify the location of crack initialization. Predict arbitrary crack propagation paths. Handle multiple cracks and the coalescence of cracks. Estimate crack width and spacing. Simulate the damage process up to failure.

Since the pioneering work by D. Ngo and A.C. Scordelis [1967] and Rashid [1968] work has been done to improve the initially presented crack concepts. With respect to the ve criteria presented above for a feasible numerical model, popular techniques available for numerical modeling of cracks within the smeared and discrete crack concepts are presented in the following sections.

17


3. Smeared vs. discrete crack modelling

3.1 Smeared crack conceptIn the smeared crack model a cracked body is represented as a continuum. This is done by smearing the eect of one or more cracks attributed to a representative volume surrounding an integration point. Smeared crack models are based on the concept of a crack band model, where the cracking strain is set equal to the crack opening, w, divided by the length of the fracture process zone, lp , see equation 3.1, Z.P. Bazant and B. Oh [1983]. The length of the fracture process zone is also referred to as the localization band. de Borst et al. [2004]

cr

=

w lp

(3.1)

The eect of cracks is translated into a stiness-deterioration in the integration point. When a combination of stresses satises a specied failure criterion, e.g. the maximum principal stress reaching the tensile strength of the concrete, cracking is initiated. Until the initiation of cracks, the concrete is modeled as an isotropic material. At the onset of cracking, the initial isotropic stress-strain law is replaced by an orthotropic law. This is done in order to represent the gradual reduction of stiness normal to the crack direction, called tension-stiening. The introduction of tension-stiening is motivated by the fact, that in reinforced concrete, the volume attributed to an integration point contains a number of cracks, and due to the bond between the concrete and reinforcement, the intact concrete between the cracks adds stiness to the model. de Borst et al. [2004] Moreover, shear stiness is added as a representation of some eects of aggregate interlock and friction within the crack. The orthotropic constitutive matrix relating stresses to strains in a two-dimensional, cracked setting is given by 3.2.

E 0 0 Ds = 0 E 0 0 0 G

(3.2)

The tension-stiening eect is represented by the parameter , which gradually decreases to zero as a function of the normal strain, nn , = ( nn ), n referring to the direction normal to the crack direction. E and G are the Young's modulus and shear modulus, respectively, and is the so-called shear retention factor representing aggregate interlock and friction within the crack. de Borst et al. [2004] The smeared crack concept suers from three major drawbacks outlined in the following. Firstly, the model is based on the concept of a crack band in which the exact location of the crack inside an element is unknown. In a case where crack width or spacing is of interest a discrete approach should be preferred over a smeared approach. Secondly, the smeared crack concept has convergence problems as a mesh-renement will aect the width of the localization band. Lastly, the strain imposed by a crack inside an element implies adjacent elements to be strained as well. This is illustrated in gure 3.1 and is referred to as stress-locking. Rots and Blaauwendraad [1989] 18

3.2. Discrete crack concept

Master Thesis

Figure 3.1.

Stress-locking is a consequence of displacement compatibility in smeared cracking. Strain of inclined crack at element 2 induced locked-in stress at element 1.

Stress-locking refers to the situation, where tensile stresses in adjacent elements is still in the elastic regime and refuses to decrease. This results in locked-in stresses at locations, such as the sides of a crack, where the stress should be zero. This drawback is a consequence of approximating a strong discontinuity using the assumption of displacement continuity. For the above mentioned reasons, a smeared crack model does not satisfy the ve specied criteria for a feasible numerical model in this report. In the following section the discrete crack concept will be introduced. Rots and Blaauwendraad [1989]

3.2 Discrete crack conceptThe discrete crack approach is the counterpart to the smeared crack approach. A crack is modeled as a geometrical discontinuity, that is, a discontinuity in displacement across the crack. In the work done by D. Ngo and A.C. Scordelis [1967], a discrete crack was initiated when the nodal force exceeded the tensile strength of the concrete. This approach suered from two drawbacks, namely: forcing the crack to propagate along element boundaries and implying a continuous change in nodal connectivity. The latter drawback refers to remeshing and the possibility that element edges do not conform to and recreate the intended crack geometry. This is especially the case for curved cracks. A more sophisticated approach to modeling of discrete cracks is the interface-technique, where interface elements are inserted along element boundaries. An example of an interface element can be seen in gure 3.2. As indicated, the thickness is almost zero. Moreover, a large dummy stiness is assigned to the interface element in order not to aect the stiness of the structure being investigated. Upon cracking the large dummy stiness is set to zero. Rots and Blaauwendraad [1989]

Figure 3.2.

Standard three-noded two-dimensional interface element. J.L. Asferg [2006]

19


3. Smeared vs. discrete crack modelling

Interface elements are based on a traction-separation description for modeling of cohesive cracks. The relation between the stress and crack opening gradients for a two-dimensional conguration are given by equation 3.3. J.L. Asferg [2006]

D11 D12 = D21 D22

n t

(3.3)

D n t

Gradient the of normal stresses acting along the interface. Gradient the of shear stresses acting along the interface. Constitutive matrix. Index 1 and 2 refer to the normal and tangential direction, respectively. Gradient the of crack opening in the normal direction. Gradient the of crack opening in the tangential direction.

Interface-elements have been used in a number of applications, e.g. C. M. Lpez [2008], where non-linear fracture is modeled for uniaxial tension loading of plain concrete. More recently, an analysis of ber reinforced polymer, FRP, strengthened reinforced concrete members has been performed, N. Khomwan, S.J. Foster and S.T. Smith [2010]. The stress transfer between the FRP and the reinforced concrete is modeled using 2-dimensional interface-elements. A bond-stress slip law was used for the description of the stress transfer at the interface. The use of interface-elements, however, puts a constraint on the crack propagation path. This is so, because the crack is constrained to propagate along the inserted interfaceelements. This constrain renders the modeling of a-priori unknown crack paths dicult. A remedy is re-meshing at every simulation stage, however the mesh must conform to the crack geometry, which in the case of curved or intersecting cracks is dicult to obtain. Several attempts have been made to construct eective remeshing algorithms, e.g. by A.R. Ingraea [1985], that reduced the mesh bias. However, such algorithms are computationally expensive. The missing possibilities of identifying the location of crack initialization and prediction of arbitrary crack propagation paths render the method unpreferable according to the ve specied criteria for a feasible numerical model in this report. The mesh-dependence was to a large extent alleviated by the advent of so-called enriched methods. An example of such enriched methods is the XFEM described in chapter 4. General for all enriched methods is to enrich the polynomial approximation space such that non-smooth solutions can be modeled independent of the mesh. This enables the class of methods to model discontinuities at arbitrary locations inside element interiors, such as a displacement discontinuity imposed by a crack. Moreover, enriched methods put no restriction on the number of cracks in the model, or whether the cracks are predened or initiated by fullling a material fracture criterion. For j cracks in the model, the polynomial approximation space is expanded to a sum over the j nodal sets describing the j cracks. Finally, by belonging to the class of discrete modeling, the XFEM is able to estimate crack width and spacing. According to the ve specied criteria for a feasible 20

3.2. Discrete crack concept

Master Thesis

numerical model in this report, the XFEM seems a valid candidate. With respect to the last point referring to the simulation of complete failure, this is a question of the numerical solver implemented in Abaqus 6.10. Fries and Belytschko. [2000] A drawback of the discrete crack concept is that it is intended for the representation of dominating cracks, thus neglecting the eect of microcracks known to occur in heterogeneous materials like concrete. However, the XFEM has attractive properties with respect to crack modeling and will be used further on in this project.

21

The eXtended Finite Element Method

4

This chapter presents the general formulation of the XFEM. Initially, the background of the XFEM is presented and an introduction to the concept of discontinuities is given. This is followed by a description of a formulation of the approximation space in the XFEM. A convenient method for choosing the set of so-called enriched nodes is presented and the concept of blending elements is introduced. Various enrichment types are discussed, followed by a description of two methods for numerical integration of the weak form. Finally, the XFEM approximation to the displacement is derived using the variational principle. Unless stated otherwise the sources used in this chapter are Fries and Belytschko. [2000] and Thomas-Peter Fries and Andreas Zilian. [2010] .Discontinuities and singularities in eld quantities are observed in many areas of civil engineering, e.g. singular stresses and strains in the vicinity of a crack-tip, or a jump in displacement across a crack. For the numerical approximation of these non-smooth variables two fundamentally dierent approaches exist. The rst method relies on polynomial approximation, based on nite element shape functions, and requires the mesh to conform to the discontinuities. Moreover, a rened mesh is required in areas where eld quantities exhibit high gradients, and remeshing is required in order to model the evolution of interfaces, e.g. cracks, boundary layers and phase transition. However, for complex geometries an eective remeshing algorithm can be dicult to construct, as the elements must conform to the geometry of the discontinuity or projection errors are introduced. Moreover, this is computationally expensive and not suited for evolving interfaces. The second, fundamentally dierent, method is based on enriching the polynomial approximation space with discontinuous functions, such that non-smooth solutions can be modeled independent of the mesh. This is a basic principle of the XFEM, and was developed by Belytschko and Black [1999] and N.Mos and Belytschko [1999], based on the partition of unity concept pioneered by Melenk and Babuska [1996]. In the following chapter, concepts behind the XFEM for treating discontinuities will be described. In order to have a common terminology for the description of discontinuities, the following section is dedicated to this cause.

23


4. The eXtended Finite Element Method

4.1 Discontinuities and high gradientsDiscontinuities are observed in the real world where eld quantities change rapidly over a length scale that is small compared to the observed domain. For a reinforced concrete beam, a jump in stress occurs across the material interface separating the concrete and the reinforcement. At the onset of cracking, stresses and strains change discontinuously across a crack and become singular at the crack-tip. Moreover, the displacement eld is discontinuous across the crack. An understanding of these phenomena is important in numerical modeling of reinforced concrete, and knowledge of the formation and propagation of cracks is vital for reliability and damage analysis of structures. For the remainder of this report, the following denition will be adopted for a discontinuity:

A discontinuity is a rapid change of a eld quantity over a length negligible in comparison to the dimensions of the observed domain.In reality a eld quantity may never change over a length of zero. However, this is justied when compared to the length scale of the observed domain. In the case where the length scale is small but has to be accounted for, the term "high gradient" will be used. The location in space, over which eld quantities or their gradients change discontinuously, will be termed interface. A mathematical description of interface is: Consider a d-dimensional domain Rd , then a manifold Rd1 is called interface. In this way, an interface is a surface in three dimensions, a line in two dimensions and a point in one dimension. Examples of two types of interfaces are given in gure 4.1.

Figure 4.1.

Example of (a) open interface and (b) closed interface.

The interfaces in gure 4.1 dier by having and not having a free end in the domain. Figure 4.1a is a so-called open interface, because the interface ends inside the domain. An example of an open interface is a crack. Figure 4.1b is a so-called closed interface, because the interface does not have any free ends inside the domain. An example of a closed interface is a material interface. The topological dierence between an open and a closed interface is described by the level-set method, described in section 4.2. A distinction is made between moving and xed interfaces. A xed interface is treated by a Lagrangian description, meaning the relative position of the interface is unchanged during deformation of the body. A moving interface is treated by an Eulerian description, 24

4.2. Level-set method

Master Thesis

meaning the interface moves through the domain. The initial position of the interface is given, and the future position is then part of the solution. In this report, the cracks appearing in the 3-dimensional concrete beam are xed interfaces. However, the crack propagates and one would assume the interface to be moving. This is not the case, since the crack propagation speed is unknown and not part of the solution in the displacement variational principle. For this reason, the propagation of the crack is treated as a quasistatic process and thus, as a xed interface. A nal distinction is made between so-called strong and weak discontinuities. Strong discontinuities refer to a jump in a eld quantity across an interface, whereas a weak discontinuity refers to a jump in the gradient of the eld quantity across an interface. A discontinuity in the gradient is also referred to as a kink in the eld variable. An example of a strong and weak discontinuities is given in gure 4.2.

Figure 4.2.

Example of (a) strong discontinuity and (b) weak discontinuity in a eld quantity represented as a surface. The interface is represented as a bold line.

With the denition of strong and weak discontinuities, the displacement exhibits a strong discontinuity across a crack and a weak discontinuity across a material interface. An accurate description of the location of the interfaces, e.g. cracks, is necessary in order to enrich the solution appropriately. This issue is addressed in the following section.

4.2 Level-set methodThe level-set method is a technique for locating interfaces and is useful in combination with the XFEM, because it facilitates the construction of the enrichment, as will be shown later. However, the method is not a part of the XFEM but is widely used in combination hereof. The level-set method denes interfaces implicitly by the zero-level of a scalar function. The method is restricted by the requirements, that the scalar function must be a continuous function and change sign across the interface. The signed distance function is a particularly useful function in this regard, because it fulls the requirements to the scalar

25



function and is easy to implement into a code. Examples of eligible level-set functions for a one-dimensional bar with a discontinuity located at x = 0 are shown in gure 4.3.

Figure 4.3.

Eligible level-set functions. The red line is the signed distance function and the black line is an arbitrary level-set function. The interface is located at the red circle on a one-dimensional bar discretized with nodes indicated as blue stars.

The signed distance function is given by equation 4.1. As the name suggests, the signed distance function computes the distance from the discontinuity to a given point and assigns a sign to the distance.

(x) = min x x , x

x

(4.1)

x .

The The The The

coordinates of a node on the interface. set of all nodes x on the interface. 2 2 2 Euclidean norm z = z1 + z2 + ... + zn . considered domain.

The sign in equation 4.1 is determined by the sign-equation sign(n (x x )), where n is the normal vector to given by n = is the dierential operator and , where 26

4.2. Level-set method

Master Thesis

= 1 holds for the signed distance function. By convention, n points from the negative subdomain into the -positive subdomain, which is the reason for the existence of the sign-equation.Because exact functional representation of discontinuities is often inconvenient, the discontinuities are stored in a discrete way. This is done by evaluating the signed distance function at the nodes, and using standard nite element shape functions to interpolate in-between, see equation 4.2. Note that an error is introduced in this approximation, which decreases with mesh renement.

h =iI

Ni (x) (xi )

(4.2)

h and x and xi i and I Ni

The approximated level-set function value and exact level-set function value. Nodal coordinate and the coordinates of node i. Node i in the set of all nodes I . Standard Finite Element shape function belonging to node i.

As previously mentioned, the topological dierence between an open and a closed interface is described by the number of level-set functions needed to describe the discontinuity. In the following two subsections, open and closed interfaces are described using the level-set method.

4.2.1 Description of closed interfacesConsider a domain Rd containing an interface. can be decomposed into two subdomains, 1 and 2 , such that = 1 2 and the interface 12 = 1 2 . 1 and 2 may consist of disconnected regions. The interface is then given by the set

12 = {x : (x) = 0}This situation is depicted in gure 4.4, where the signed-distance function has been used as the level-set function.

27



Figure 4.4.

(a) The domain is decomposed into two subdomains 1 and 2 . 12 describes the interface. The normal vector, n, points from the -negative subdomain into the -positive subdomain. (b) Contour values of the signed-distance function.

For more than 2 subdomains, 1 level-set function is no longer sucient. In general, for closed interfaces, n level-set functions can separate 2n subdomains.

4.2.2 Description of open interfacesConsider a domain Rd partially cut by an interface. Where one level-set function is able to describe a closed interface, the description of an open interface requires a second level-set function, , to describe where the interface ends. The interface is then given by the set

12 = {x : (x) = 0 and (x) 0} has to fulll the same requirements as those put on . However, in computational implementations, is often chosen a straight line orthogonal to the tip of the interface and is the signed-distance function, which is zero across the interface and extended tangentially from the crack tip. In other words, is not necessarily a signed-distance function, but used to dene the end of the open interface, and describes a closed interface. In Abaqus 6.10, and are both signed-distance functions. A crack is a typical open interface, and an example is depicted in gure 4.5.

28

4.3. General formulation of the XFEM

Master Thesis

Figure 4.5.

(a) The domain partially cut by a crack. (b) The signed-distance function discribing the crack path. (c) The level-set function dening the crack tips.

In the following section an XFEM approximation of the displacement is presented.

4.3 General formulation of the XFEMThe XFEM is a numerical method that enables the local enrichment of approximation spaces by including known solution properties into the approximation space. Consider an n -dimensional domain Rn , which is discretized by nel elements, numbered from 1 to nel . I is the set of all nodes. The general formulation of the standard XFEM for the approximation of the unknown displacement u(x) is of the form shown in equation 4.3.

uh (x) = iI

Ni (x) ui + i I1 Strd. F EM approx.

Mi1 (x) a1 + . . . + i i Im Enrichment 1

Mim (x) am iEnrichment m

(4.3)

29

Group B122b - Spring 2011uh (x) Ni (x) ui I Mim (x) am i Im


The approximation of the displacement. Standard FEM shape function of node i. The degree of freedom of the standard FEM part at node i. The set of all nodes. The local enrichment function of node i belonging to the m 'th enrichment. The degree of freedom of the m 'th enrichment of node i. The nodal subset of the m 'th enrichment, Im I .

Note that the approximation in 4.3 consists of a standard nite element part plus additional m enrichment terms. This form of enrichment is called extrinsic enrichment, because special enrichment terms are added to the polynomial approximation space, resulting in more functions and nodal unknowns to be evaluated. The alternative to extrinsic enrichment is intrinsic enrichment, where the standard nite element shape functions are replaced by special shape functions, which are able to capture the nonsmooth solution. This is done by expanding the function basis for the elements cut by a discontinuity, and results in no additional unknowns. However, the amount of computational work needed to establish the special shape functions makes intrinsic enrichment unappealing in comparison to extrinsic enrichment. In Abaqus 6.10 extrinsic enrichment is used. Each enrichment consists of a local enrichment function, Mim (x), and additional nodal unknowns am describing the character of the m'th discontinuity, e.g. if the m'th i discontinuity is a crack, the local enrichment function reects the inuence of the crack by introducing a jump in the displacement eld. The local enrichment function is given by equation 4.4.

Mim (x) = Ni (x) m (x) Ni (x) m (x)Partition of unity function of node i. Global enrichment function of the m 'th enrichment.

(4.4)

The global enrichment function, m (x), incorporates the special knowledge of the solution properties into the approximation space. The partition of unity functions, Ni (x), only build a partition of unity in a local part of the domain, that is, in elements whose nodes are all in the nodal subset I . In elements that are fully enriched the property of 4.5 holds, that is, Ni (x) build a partition of unity.

Ni (x) = 1, iIj

x , j = 1, . . . , m. j

(4.5)

The partition of unity concept is a well-known property of the standard nite element shape functions. For this reason Ni (x) = Ni (x) is an option, but not a necessity. For 30

4.4. Choice of enriched nodes

Master Thesis

example, Ni (x) can be chosen quadratic while using linear standard nite element shape functions, Ni (x). In Abaqus 6.10 Ni (x) = Ni (x). The approximation on the form shown in 4.3 does not have the Kronecker- property. This is seen by evaluating the approximation at x = xk

uh (xk ) = iI

Ni (xk ) uk + i I1

Mi1 (xk ) a1 + . . . + k i Im

Mim (xk ) am k

(4.6)

which for i = k yields

uh (xk ) = uk + i I1

Mi1 (xk ) a1 + . . . + k i Im

Mim (xk ) am k

(4.7)

Consequently, uh (xk ) = uk , which renders the imposition of essential boundary conditions dicult and the computed unknowns are no longer the sought functions values. It is therefore desirable to recover the Kronecker- property, which is achieved by making the enrichment terms vanish at the nodes. This is done by shifting the approximation. The shifted approximation of the displacement is shown in 4.8. Note, that the expression for the local enrichment function 4.4, has been inserted. For brevity in the expression, only one enrichment term has been included.

uh (x) = iI

Ni (x) ui + i I1

Ni (x) [(x) (xi )] ai

(4.8)

By evaluating 4.8 in x = xk it can be shown, that the Kronecker- property is recovered. Abaqus 6.10 is using the shifted form of the approximation. The following section will describe a method for choosing the nodal subset I .

4.4 Choice of enriched nodesFor computational eciency, partition of unity enrichments are preferably localized to the sub-domains where they are needed. In other words, only a nodal subset needs enrichment. By enrichment the authors allude to including enrichment terms in the approximation of the displacement. The nodal subset I is built from the nodes of the elements cut by a discontinuity. A convenient method for choosing the enriched nodes is the level-set method described in section 4.2. The level-set function determines whether or not an element is cut by a discontinuity. The signed distance function is used for determining whether an element is cut or not:

31



Cut element: Uncut element:

iI el iI el

min (sign((xi ))) max (sign((xi ))) < 0iI el iI el

min (sign((xi ))) max (sign((xi ))) > 0

I el is the set of element nodes. In other words, an element is cut if the level-set function changes sign in the element.For identifying an element containing a crack-tip the following two criteria of the signed distance function must be met simultaneously:

iI el

min (sign((xi ))) max (sign((xi ))) < 0iI el iI el

min (sign((xi ))) max (sign((xi ))) < 0iI el

The reason for identifying the crack tip element is to properly enrich it so that stress and strain singularities are accounted for. However, if only one element is enriched sub-optimal convergence rates are obtained. This is due to the fact, that mesh renement reduces the area over which the singularity is accounted for. An alternative method that accounts for this problem is the branch enrichment approach, where nodes within a certain radius from the crack tip are enriched. In this way the enriched area is kept constant during mesh renement.

Itip = {i : xi xi < r}

where xi xi is the Euclidean distance between a point, xi , and the crack-tip, xi . Abaqus 6.10 uses this approach, for static cracks, with an enrichment radius of three times the characteristic element length. For evolving cracks the crack tip enrichment has not been implemented in Abaqus 6.10. The mathematical description of the enrichment functions used in Abaqus 6.10 are described in section 4.5. Since only a subset of the nodes are enriched three types of elements can be dened: 1. A standard element if none of its nodes are enriched. 2. A fully enriched element if all of its nodes are enriched. 3. A partly enriched element if some of its nodes are enriched. Figure 4.6 illustrates a one-dimensional and a two-dimensional example of the three element types and Ni (x), here chosen as a linear function.

32

4.5. Global enrichment functions

Master Thesis

Figure 4.6.

Discretized domains in one- and two dimensions with nodal subset I . (a) and (c) show the reproducing-, blending- and standard elements. (b) and (d) show that the function Ni (x), here linear, only builds a partition of unity in reproducing elements and varies linearly from one to zero over the blending element.

The fully enriched elements are called reproducing elements, because the approximation shown in 4.3 is able to reproduce any enrichment function exactly in . The partly enriched elements are also referred to as blending elements because the enrichment is blended over the element. This is because the partition of unity functions do not build a partition of unity, which introduces parasitic terms into the approximation if linear or higher order global enrichment functions are chosen. In Abaqus 6.10 blending elements do not exist, because the meshing algorithm is constructed such that element boundaries conform to interfaces. Moreover, only discontinuous enrichment functions, with constant variation, are used, which eectively eliminates potential problems caused by blending elements.

4.5 Global enrichment functionsAs previously mentioned distinction is made between weak and strong discontinuities. Moreover, singularities in the stress and strain eld near a crack tip must be reected in the solution. In the following the global enrichment functions, used to reect these phenomena in the solution, are presented.

33



4.5.1 Weak discontinuitiesA weak discontinuity refers to a kink in the solution, that is, a jump in the gradient of the solution. A well-known example of a weak discontinuity is at a material interface, where stresses or strains change discontinuously across the material interface. For weak discontinuities one choice for the global enrichment function is the abs-function, which is shown in 4.9. Note that the abs-function, also referred to as abs-enrichment, uses the levelset function. As previously mentioned, the level-set function is convenient in corporation with the XFEM, because it nds use in the construction of the global enrichment function.

(x) = abs((x)) = |(x)|The gradient of the abs-function is given in equation 4.10.

(4.9)

(x) = sign((x))

(x)

(4.10)

However, this type of enrichment function leads to trouble in blending elements. This is because the function has a linear variation in the domain . As previously mentioned, troubles arise because the partition of unity function, N (x), does not build a partition of unity in the blending elements. When (x) is multiplied with N (x), a parasitic term is introduced into the approximation. However, this is only the case if the order of N (x) and N (x) is the same. For example, if both shape functions are chosen linear, the enrichment term becomes parabolic because of the linear variation of (x) and thus, a linear term is summed with a parabolic term in the blending elements. In other words, the standard FEM part cannot compensate for the error introduced by the parasitic term. A remedy for this problem is to choose N (x) one order higher than N (x), in which case the standard FEM part will be able to compensate for the error introduced in the approximation by the parasitic term. An improvement to the abs-function was introduced by N.Mos and Belytschko [1999]. The improved function is the so-called modied abs-enrichment, which has the property of being non-zero only in the fully enriched elements. By being zero in the blending elements, no parasitic terms are introduced into the approximation and thus optimal convergence rates can be obtained. The modied abs-enrichemt is shown in 4.11.

(x) =iI

|i | Ni (x) iI

i Ni (x)

(4.11)

Referring to the problem at hand in this project, a reinforced concrete beam, the reinforcement and the beam are meshed as independent parts in Abaqus 6.10. This entails that no elements contain more than one material property and thus, no weak discontinuities are present in the model because the material interface is coincident with the element boundaries. In the following section two global enrichment functions for strong discontinuities will be presented. 34

4.5. Global enrichment functions

Master Thesis

4.5.2 Strong discontinuitiesA strong discontinuity refers to a jump in the solution. Two popular choices for a global enrichment function are the Heaviside function and the sign-function. Both functions utilize the level-set function, again proving its usefulness in the context of the XFEM. Although dierent in structure, the two functions yield identical results as they span the same approximation space. The Heaviside function and the sign-function are shown in equations 4.12 and 4.13, respectively.

(x) = H((x)) =

0 1

: (x) 0 : (x) > 0

(4.12)

(x) = sign((x)) =

1 0 1

: (x) < 0 : (x) = 0 : (x) > 0(4.13)

The gradient of these enrichment functions is zero. Note, that the functions do not cause trouble in blending elements, because they are constant in . In Abaqus 6.10 the jumpfunction in 4.14 is used.

(x) = H((x)) =

1 1

: (x) < 0 : (x) 0

(4.14)

4.5.3 SingularitiesAt the crack tip a global enrichment function with a singular derivative is needed. Moreover, the function must be discontinuous along the crack. In practice the four global enrichment functions in equations 4.15 to 4.18 are often used.

2 2 (x) = r sin sin 2 3 (x) = r cos 2 4 (x) = r cos sin 2 1 (x) = r sin

(4.15) (4.16) (4.17) (4.18)

The functions depend on a local polar coordinate system at the crack-tip, see gure 4.7, where = 0 is tangent at the crack-tip.

35



Figure 4.7.

The polar coordinate system around the crack-tip. xtip and ytip are the Cartesian coordinates of the crack-tip. (r, ) are the radial and angular coordinates, respectively, from the pole to a point, (x, y).

The four global enrichment functions in equations 4.15 to 4.18 are a result of linear elastic fracture mechanics, LEFM. They span the linear asymptotic crack-tip function of elasto-statics, and 4.15 takes the discontinuity in displacement into account. These are important in crack modeling, because if only sign-enrichment was used, the crack would be virtually extended to the boundary of the element in which the crack-tip is present. Using crack-tip enrichment functions ensures that the crack ends exactly at the location of the crack-tip. Moreover, 1 (x) to 4 (x) are an analytical result from LEFM to the near tip behavior, that is, the accuracy of the approximation is increased by including analytical results in the approximation. Abaqus 6.10 takes advantage of these four functions in representing the singular stress and strain eld near singularities. However, this is only the case for stationary cracks, because accurate modeling of the crack-tip singularity requires constantly keeping track of the crack location, and the degree of the singularity depends on the location in non-isotropic material, e.g. concrete. Moving cracks are modeled with the so-called cohesive segments method and phantom nodes. This matter is addressed in the following.

4.6 Cohesive segments methodThe XFEM-based cohesive segments method can be used to simulate crack initiation and propagation along an arbitrary, solution-dependent path. The cohesive segments method is based on the insertion of a cohesive segment through an element once a decohesion criteria is met, e.g. a damage criteria. The segments are not, as previously described, restricted to being located along element boundaries, but can be located at arbitrary locations and in arbitrary directions, allowing for the resolution of complex crack patterns. For this reason, a cohesive segment is not to be mistaken for an interface element, described in chapter 3. The segment is taken to extend through the element to the boundary in which it is inserted. The method is based on the partition of unity property of nite element shape functions and enriching the approximation space with discontinuous functions, as 36

4.7. Phantom-node method

Master Thesis

described in the previous sections. The displacement jump across a crack is described using the Phantom node method in Abaqus 6.10.

4.7 Phantom-node methodThis section is based on the sources M.J McNary [2009], T.Rabczuk et al. [2008], Song et al. [2006] Equivalent to the XFEM it is possible to model discontinuities at arbitrary locations in the mesh using the so-called Phantom-node method. The Phantom-node method is based on adding an extra element on top of an existing, cracked element. In contrast to the XFEM the crack kinematics is obtained by overlapping elements instead of introducing additional degrees of freedom. Abaqus 6.10 has a limitation of 20 degrees of freedom available per node. This limit is exceeded in special cases when using the XFEM, which has prompted Dassault Systmes Simulia Corp. [2010] to use the Phantom node method in its implementation, which is an alternative approach within the framework of the XFEM. Figure 4.8 shows an example of a cracked domain 0 , supported along the boundary u and subjected to a traction, t, applied along the boundary t .

Figure 4.8.

The principle of the Phantom-node method. On top of the nodes in the cracked elements phantom nodes are added and the integration is performed over the hatched area. Solid circles represent real nodes and hollow circles represent phantom nodes. f (X) is the signed-distance level-set function evaluated at the Cartesian coordinate X .

37



In the cracked elements phantom nodes are added on top of the real nodes and thus, leading to an additional element on top of the cracked element. Each element consists of a real subdomain and a phantom subdomain, e.g. + and , where 0 = + with p p 0 0 reference to gure 4.8. Then the displacement eld in the real domain can be interpolated using the degrees of freedom in the phantom domain. Initially the real node and the phantom node are tied together. When cracking occurs, e.g. by fullling a damage criterion, each phantom node and its corresponding real node are no longer tied together and can move apart. The magnitude of the separation, i.e. the crack opening, is governed by a cohesive law until the cohesive strength of the element is zero. This relation is governed by the cohesive segments method. In Abaqus 6.10 the relation between crack opening and traction is linear and given as input to the model by prescribing a fracture energy. This is described in detail in chapter 5. While the XFEM is suitable for modeling the singular stresses and strains around a cracktip, the Phantom-node method is only applicable to cohesive crack modeling. In this fashion, the crack is extended to the element boundary and the singular eld is replaced by a cohesive traction. However, the Phantom-node method has been extended to model the crack-tip inside an element by T. Rabczuk and Wall [2008]. Abaqus 6.10 uses the former approach, that is, the crack-tip always ends at an element boundary. This is a simplied approach to crack modeling, as mesh-sensitivity is introduced for crude mesh densities. For this reason the possibility of a precise evaluation of the crack-tip location and propagation is limited. Figure 4.9 illustrates the dierence between the XFEM and the Phantom-node method for a one-dimensional bar with an inter-element discontinuity.

Figure 4.9.

The interpolation basis of the XFEM vs. the Phantom-node method for a onedimensional element. T.Rabczuk et al. [2008]

38

4.8. Numerical integration of the weak form

Master Thesis

Note that the displacement jump, [[u]], in gure 4.9 is identical for the two methods. In the Phantom-node method the two elements representing the cracked element do not share nodes and therefore have independent displacement elds. Both elements are only partially active, which is represented numerically in the denition of the displacement eld by introducing the jump function, see equation 4.14, which is active based on the signed-distance level-set function. The displacement jump over a crack is then dened as the dierence between the displacement elds of the two elements. The approximation of the displacement, using the Phantom-node method, is given in equation 4.19.

uh (X) =+ I {w0 , wp }

NI (x) uI H(f (X)) + + J {w0 , wp }

NJ (x) uJ H(f (X))(4.19)

+ + where w0 , w0 , wp and wp are the nodes belonging to + , , + and , respectively. It p p 0 0 has been shown that the Phantom-node method is equivalent to the XFEM. An example is the similarity between enriched degrees of freedom in the XFEM and the phantom degrees of freedom in the Phantom-node method. From a computational point of view, the Phantom-node method is superior to the XFEM, however more simple for previously mentioned reasons, e.g. crack-tip position. Note, that the Phantom-node method is based on the XFEM, in the sense that the standard FEM approximation space is enriched. Moreover, both methods reect the jump in the displacement eld by piecewise integration. Furthermore, both methods use the level-set method in the topological description of the discontinuity and in the evaluation of the global enrichment functions. The numerical integration method used in both methods is described in the following section.

4.8 Numerical integration of the weak formStrong discontinuities will be present in the reinforced concrete beam treated in this report, that is, a jump in the displacement eld across the cracked elements. This complicates the numerical integration in the cracked elements, because standard Gauss quadrature requires the integrand to be polynomial. Within the nite element framework, this corresponds to the integration of element-wise smooth, continuous shape functions. If the numerical quadrature of the weak form is not performed correctly, the advantage of including local enrichment functions is lost. Two remedies for the numerical quadrature of the weak form exist: 1. Element decomposition. 2. Integrand transformation. Element decomposition is a widely used technique, because it is computationally less demanding than the integrand transformation. Only element decomposition is described in the following, because Abaqus 6.10 utilizes this technique. 39



Element decomposition refers to a sub-division into smaller elements in the element containing the discontinuity. The sub-division is carried out in the reference element, that is, the element described in the isoparametric space. Isoparametric mapping is used to map the reference element from the isoparametric space to the physical space. Figure 4.10 shows a triangular element cut by a crack. The triangle is decomposed into two dierent elements: A smaller triangle and a quadrilateral, both aligning with the geometry of the crack.

Figure 4.10.

Element decomposition of a triangle into two sub-elements: A triangle and a quadrilateral. Four Gauss points are placed in each sub-element.

Standard Gauss quadrature is then performed over the sub-elements, in which the integrand is now smooth. Note that the sub-elements each have been assigned a number of Gauss points, in this case four. The number of Gauss points and the type of sub-elements used in Abaqus 6.10 are unknown to the authors. The crack is then an internal boundary of the domain of integration. A mathematical description of the integration is given in equation 4.20.K k=1 k e

f (x)non-smooth dx =e

f (x)smooth dx

(4.20)

The sub-division is done by non-overlapping elements that must conform to the same requirement as the continuous problem. Note that element sub-division is not equivalent to remeshing, as no additional degrees of freedom are introduced. The sub-elements are only introduced for the purpose of integration. Moreover, the basis function is associated with the nodes of the parent domain, which implies that no restriction is imposed on the shape of the sub-elements. Recall, that the discontinuity is dened as the zero-level of the interpolated level-set function, see equation 4.2.

h =iI

Ni (x) (xi ) = 0

40

4.9. Governing equations

Master Thesis

Dierent shape functions can be used for the interpolation than for the approximation of the displacement. Linear shape functions are particularly useful, as the discontinuity is a straight line in the reference element, and also when mapped in the real element. This simplies the integration signicantly, as an element sub-division algorithm is more easily constructed, than for a curved discontinuity. However, linear shape functions only allows for a piecewise linear representation of discontinuities. Bi-linear shape functions produce curved discontinuities, and an ecient element sub-division algorithm is dicult to construct. For this reason, if bi-linear or higher order shape functions are used, linearization of the discontinuity is often used. The discontinuity is linearized by drawing a straight line between the points on the element edges that are cut by the discontinuity, see gure 4.11.

Figure 4.11.

(a) A curved interface in a bi-linear element can be (b) linearized by neglecting the curvature of the interface or (c) the element is decomposed into two triangles and linear interpolation is assumed.

The nodal values of the level set function are interpolated using the same shape functions as in the approximation of the displacement. However, it is unknown whether a linearization of the interface is performed. In either case, when the sub-division has been performed, standard Gauss quadrature is adopted. The element decomposition approach is favorable in computational implementation, because existing nite element integration schemes do not need any modication. In the following section the weak form of the equilibrium equations are presented. Systmes [2010]

4.9 Governing equationsThe weak form of the equilibrium equations, the displacement u being the primary variable, can be stated by the principle of virtual work, see equation 4.21.

T

d =

uT b d +t

uT t dt ,

u

(4.21)

41

Group B122b - Spring 2011 t b tThe domain of integration. The traction surface. Strain tensor in vector form. Stress tensor in vector form. Prescribed body force vector. Prescribed traction vector.


where superscript T refers to the transpose. Prescribed displacements u are imposed on u , while tractions t are imposed on t . The internal boundary of the crack, c , is assumed to be traction-free. The domain is bounded by = t c u , as shown in gure 4.12.

Figure 4.12.

Domain supported on u and loaded on t . An internal crack is dened along the boundary c .

n = t on t n = 0 on c u = u on u The two-dimensional nite element discretized form of the weak form is stated in 4.22.

fx f y B T Dep B d u = d + 0 0

tx t y d K u = f + ft 0 0

(4.22)

where fx = Nstd bx and fy = Nstd by . bx and by are the body forces in the horizontal and vertical direction, respectively. ft contains the tractions tx and ty in the horizontal 42

4.9. Governing equations

Master Thesis

and vertical direction, respectively. Dep is the elasto-plastic constitutive matrix chosen according to the adopted material model of the concrete, see chapter 5. The strain distribution matrix B is given by equation 4.23.

T T x Nstd x Nenr T T B= y Nstd y Nenr T T T T y Nstd x Nstd y Nenr x Nenr

(4.23)

T T where Nstd are standard FEM shape functions, and Nenr are the local enrichment functions given by equation 4.4. The strain distribution matrix is of dimension 3 2 (nel + n ), and el nel and n are the numbers of element nodes and enriched element nodes, respectively. el

43

Discontinuous modeling in Abaqus

5

This chapter presents a summary of the methods described in chapters 2-4, used in Abaqus 6.10 in relation to discontinuous modeling. With regard to the methods of discontinuous modeling, preliminary choices made in Abaqus 6.10 for this report regarding the element type used for the numerical discretization and the material properties of the examined concrete and steel are presented. The choices are common for the benchmark tests considering concrete and the reinforced concrete beam considered in the problem formulation of this project.In Abaqus 6.10 cracking in concrete and steel is modeled in a discrete fashion, that is, a strong discontinuity is introduced in the displacement eld by the Phantom-node method, in which the displacement jump is reproduced by introducing the jump function. For the remainder of this report the Phantom-node method will be referred to as the XFEM because the methods are equivalent. The crack is represented as an open interface, but since the crack is always extended to the boundary of the element, in which it is present, only one level-set function is used for the topological description of a crack. The levelset function is chosen as the signed-distance function and is interpolated using the same interpolation functions as the approximation of the displacement. The crack is modeled by inserting a cohesive segment in the cracked element. The cohesive segments method is based on the ctitious crack model by A. Hillerborg and Peterson [1976], that is, nonlinear fracture mechanics is used. The adopted crack initiation criterion is the maximum principal stress criterion, in which a crack is initiated if the maximum principal stress reaches the tensile strength of the concrete. The crack propagates perpendicular to the direction of the maximum principal stress. The evolution of the crack is governed by the fracture energy, which represents the tension-softening behavior of the concrete during cracking. The relationship between the crack opening displacement and the closing stresses acting on the crack is linear, in which case the critical crack opening displacement, wc , is

1/2 ft wmax = Gf wmax = 2 Gf /ftwhich is found by simple geometrical considerations, see gure 5.1.

(5.1)

45


5. Discontinuous modeling in Abaqus

Figure 5.1.

Linear relationship between the closing stress and crack opening displacement upon cracking.

The modeling of cracked concrete is illustrated in gure 5.2.

Figure 5.2.

The modeling concept used in Abaqus 6.10 to model cracks in concrete.

The constitutive behavior of concrete is modeled using the so-called Concrete Damaged Plasticity material model, abbreviated CDP. In tension the CDP material model is used in coordination with the XFEM and the cohesive segments method, that is, CDP is used until tensile crack initiation is detected, at which point a cohesive segment is inserted and the XFEM is activated. For compression the CDP material model is used without the XFEM. The CDP material model is a continuum-damage based constitutive relation. The stress-strain relation is given by equation 5.2.

el = (1 d)D0 : (

pl

) = Del : (

pl

)

(5.2)

46

Master Thesisel D0 Del d

Initial, undamaged elastic stiness matrix of concrete. Degraded elastic stiness matrix of concrete. Scalar degradation variable.

The scalar degradation variable has an initial value of zero for intact material and increases towards one for complete loss of material stiness. Based on the observation that concrete is anisotropic, the scalar degradation variable is dierent in tension and compression. Since the scalar degradation variable is not used for tension, it will not be further discussed in this

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XFEM for reinforced concrete